The Compensation Method of Vehicle Magnetic Interference for the Magnetic Gradiometer

Abstract

The magnetic interference of vehicle imposes a strong influence on the magnetic gradiometer. Based on the mechanism of the vehicle magnetic interference, we firstly use the difference algorithm of the magnetic gradient tensor to fuse the magnetic interference of each vector magnetometer and establish a mathematical model of vehicle magnetic interference for the magnetic gradiometer. Next, we propose a compensation method for the vehicle magnetic interference and a recognition method for the estimation of compensation coefficients based on this mathematical model. The simulation results show that the proposed method can compensate as much as 96.2% of the vehicle magnetic interference efficiently.

1. Introduction

The magnetic gradiometer onboard Unmanned Underwater Vehicle (UUV) for underwater object detection is attracting particular interest in recent years for its advantages in remote control and autonomous operation. The magnetic gradiometer is constructed by three spatial derivatives along orthogonal directions and a magnetic gradient is measured as the difference between two magnetic readings at different locations. The magnetic gradiometer formed by fluxgate magnetometers has several advantages such as small size and low power with relatively high sensitivity. Gradiometer has also the advantage of rejecting unwanted long-range environmental noise sources. Various critical techniques have been developed and demonstrated [1–3].

It is inevitable that the magnetic gradiometer is influenced by the magnetic interference of vehicle, which mainly includes the induced magnetic field and the permanent magnetic field of vehicle [4–7]. So it is necessary to compensate this interference. Bono et al. studied on magnetic interference of the magnetic gradiometer onboard the UUV and proposed a scheme for the compensation of vehicle magnetic interference [8]. Keene et al. developed a Superconductivity Quantum Interference Device (SQUID) gradiometer system that used active shielding and an adaptive signal processing algorithm to eliminate magnetic interference [9], but he did not expound the compensation algorithm of vehicle magnetic interference. Pei and Yeo developed a prototype magnetic gradiometer onboard UUV for magnetic detection of underwater objects, which was housed in a plastic nose cone consisting of four fluxgate magnetometers, and one of the magnetometers served as the reference reading to the earth magnetic fields and was used to compensate the platform magnetic interference of other magnetometers [10, 11]. However this method did not consider the platform interference of the reference magnetometer.

The objective of this paper is to compensate vehicle magnetic interference for the magnetic gradiometer; we firstly propose a mathematical model of vehicle magnetic interference by using the difference algorithm of the magnetic gradient tensor to fuse the magnetic interference of each vector magnetometer and then we propose a compensation method for the vehicle magnetic interference and a recognition method for the estimation of the compensation coefficients. This method uses one of the magnetometers to serve as the reference and take into consideration the vehicle magnetic interference, so it can compensate the vehicle magnetic interference of the magnetic gradiometer theoretically. Finally, numerical simulations are carried out to show the effectiveness of the proposed compensation method.

2. Preliminaries of Magnetic Gradient Tensor

The magnetic gradient tensor is space variation rate in the 𝑋, 𝑌, 𝑍 directions of total magnetic field 𝐁(𝐵_𝑥,𝐵_𝑦,𝐵_𝑧), which is defined as 𝐆=∇𝐁=𝜕𝐵_𝑥𝜕𝑥𝜕𝐵_𝑥𝜕𝑦𝜕𝐵_𝑥𝜕𝑧𝜕𝐵_𝑦𝜕𝑥𝜕𝐵_𝑦𝜕𝑦𝜕𝐵_𝑦𝜕𝑧𝜕𝐵_𝑧𝜕𝑥𝜕𝐵_𝑧𝜕𝑦𝜕𝐵_𝑧𝜕𝑧. (1)

In general, the curl and the divergence of the magnetic field vanish outside the magnetic sources. So the relations of the nine gradients are expressed as ∇⋅𝐁=𝜕𝐵_𝑥+𝜕𝑥𝜕𝐵_𝑦+𝜕𝑦𝜕𝐵_𝑧𝜕𝜕𝑧=0,∇×𝐁=𝑖𝑗𝑘𝜕𝜕𝑥𝜕𝜕𝑦𝐵𝜕𝑧_𝑥𝐵_𝑦𝐵_𝑧=0. (2)

The magnetic gradiometer is composed of four vector magnetometers and their observations are 𝐁_𝑖(𝐵_𝑥𝑖,𝐵_𝑦𝑖,𝐵_𝑧𝑖)(𝑖=1,2,3,4), which is described by 𝐵_𝑥21𝐵_𝑥31𝐵_𝑥41=𝑑𝑥₂₁𝑑𝑦₂₁𝑑𝑧₂₁𝑑𝑥₃₁𝑑𝑦₃₁𝑑𝑧₃₁𝑑𝑥₄₁𝑑𝑦₄₁𝑑𝑧₄₁𝜕𝐵_𝑥𝜕𝑥𝜕𝐵_𝑥𝜕𝑦𝜕𝐵_𝑥𝜕𝑧, (3) where 𝐵_𝑥𝑖1 is the difference between the 𝑥 components of the 𝑖th and the first magnetometer and (𝑑𝑥_𝑖1𝑑𝑦_𝑖1𝑑𝑧_𝑖1) is the distance between the 𝑖th and the first magnetometer [12].

Suppose Δ𝐁=(𝐁₂₁𝐁₃₁𝐁₄₁), 𝐁_𝑖1=𝐁_𝑖−𝐁₁. From (3), we derive Δ𝐁^𝑇=𝐀𝐆^𝑇, (4) where the matrix 𝐀 is the distance matrix of (3). The magnetic gradient tensor 𝐆 is expressed as 𝐆=Δ𝐁𝐀^𝑇−1=Δ𝐁⋅𝐚. (5)

Equation (5) is the difference algorithm of the magnetic gradient tensor.

3. Mathematical Model of Vehicle Magnetic Interference

In the process of vehicle movement, the material of vehicle can generate much magnetic interference, such as permanent field and induced field, which can impose influence on the measurement accuracy of magnetometer. The permanent field of vehicle is generated by the magnetization of the hard magnetic material. Because the hard magnetic material has high coercivity, the permanent field can be kept for a long time. Since the magnetometer and the hard magnetic material are fixedly connected on the vehicle, no matter how the vehicle posture changes, the vector magnetometer measurement of the permanent magnetic field is a constant. The induced magnetic field is generated by the magnetization of the soft magnetic material of the vehicle. The soft magnetic material has low coercivity, so the induced magnetic field changes with the external magnetic field [13, 14]. The magnitude and direction of the induced magnetic field are associated with the posture and position of the vehicle. Three axes of the vector magnetometer are denoted as 𝑥-, 𝑦-, and 𝑧-axes of the coordinate system of vehicle. The permanent magnetic field of vehicle can be shown as 𝐁^ℎ=(𝐵^ℎ_𝑥𝐵^ℎ_𝑦𝐵^ℎ_𝑧). Based on the Poisson equation, the induced magnetic field of vehicle 𝐁^𝑠=(𝐵^𝑠_𝑥𝐵^𝑠_𝑦𝐵^𝑠_𝑧) is expressed as 𝐁^𝑠=𝐊𝐁^𝑏, (6) where 𝐁^𝑏=(𝐵^𝑏_𝑥𝐵^𝑏_𝑦𝐵^𝑏_𝑧) is the background field, 𝐊_3×3 is the induced coefficient matrix of the vehicle, and 𝐊_𝑖𝑗=𝐊_𝑗𝑖(𝑖,𝑗=1,2,3).

By analyzing the permanent and induced magnetic field, the observation of the vector magnetometer is expressed as 𝐁^′=𝐁+𝐊𝐁^𝑏+𝐁^ℎ, (7) where 𝐁 is the actual magnetic field. The observation of the 𝑖th vector magnetometer of the magnetic gradiometer can be obtained from (7): 𝐁^′_𝑖=𝐁_𝑖+𝐊_𝑖𝐁^𝑏+𝐁^ℎ_𝑖, (8) where 𝐊_𝑖 is the induced coefficient matrix of the 𝑖th vector magnetometer and 𝐁^ℎ_𝑖 is the permanent magnetic field of the 𝑖th vector magnetometer. So the difference between the observation of the 𝑖th and the first magnetometer is expressed as 𝐁^′_𝑖1=𝐁^′_𝑖−𝐁^′₁=𝐁_𝑖1+𝐊_𝑖1𝐁^𝑏+𝐁^ℎ_𝑖1, (9) where 𝐊_𝑖1=𝐊_𝑖−𝐊₁,𝐁^ℎ_𝑖1=𝐁^ℎ_𝑖−𝐁^ℎ₁. (10)

Based on the difference algorithm of the magnetic gradient tensor (5), the observation of the magnetic gradiometer 𝐆_𝑚 is expressed as 𝐆_𝑚=𝐁^′₂₁𝐁^′₃₁𝐁^′₄₁=𝐁⋅𝐚₂₁𝐁₃₁𝐁₄₁⋅𝐚+𝐊₂₁𝐁^𝑏𝐊₃₁𝐁^𝑏𝐊₄₁𝐁^𝑏+𝐁⋅𝐚_ℎ21𝐁^ℎ₃₁𝐁^ℎ₄₁𝐊⋅𝐚=𝐆+₂₁𝐁^𝑏𝐊₃₁𝐁^𝑏𝐊₄₁𝐁^𝑏+𝐁⋅𝐚_ℎ21𝐁^ℎ₃₁𝐁^ℎ₄₁⋅𝐚, (11) where 𝐆 is the actual data of the magnetic gradient tensor. And the vehicle magnetic interference 𝐃 is expressed as 𝐃=𝐆_𝑚−𝐆=𝐊₂₁𝐁^𝑏𝐊₃₁𝐁^𝑏𝐊₄₁𝐁^𝑏+𝐁⋅𝐚_ℎ21𝐁^ℎ₃₁𝐁^ℎ₄₁⋅𝐚. (12)

Equation (12) is the mathematical model of the vehicle magnetic interference.

4. Compensation Method of Vehicle Magnetic Interference

One of the magnetometers with the maximum distance from the UUV magnetic interference can serve as the reference reading to the background field 𝐁^𝑏 [10, 11]. But the reference magnetometer is also influenced by vehicle magnetic interference. We take into consideration the interference of the reference magnetometer, and the background field 𝐁^𝑏 can be expressed by the actual magnetic field of the reference: 𝐁^𝑏=𝐁₁=𝐈_3×3+𝐊₁⁻¹𝐁^′₁−𝐁^ℎ₁. (13)

The first magnetometer serves as the reference in (13). Substituting (13) into (9), 𝐁^′_𝑖1 can be expressed as 𝐁^′_𝑖1=𝐁_𝑖1+𝐊_𝑖1𝐈_3×3+𝐊₁⁻¹𝐁^′₁−𝐊_𝑖1𝐈_3×3+𝐊₁⁻¹𝐁^ℎ₁+𝐁^ℎ_𝑖1=𝐁_𝑖1+𝐤_𝑖1𝐁^′₁+𝐛^ℎ_𝑖1,𝐤_𝑖1=𝐊_𝑖1𝐈_3×3+𝐊₁⁻¹,𝐛^ℎ_𝑖1=𝐁^ℎ_𝑖1−𝐊_𝑖1𝐈_3×3+𝐊₁⁻¹𝐁^ℎ₁, (14) where 𝐤_𝑖1 and 𝐛^ℎ_𝑖1 are constant matrices. The observation of the magnetic gradient tensor can be expressed as 𝐆_𝑚=𝐁^′₂₁𝐁^′₃₁𝐁^′₄₁=𝐁⋅𝐚₂₁𝐁₃₁𝐁₄₁⋅𝐚+𝐤₂₁𝐁^′₁𝐤₃₁𝐁^′₁𝐤₄₁𝐁^′₁+𝐛⋅𝐚_ℎ21𝐛^ℎ₃₁𝐛^ℎ₄₁𝐤⋅𝐚=𝐆+₂₁𝐁^′₁𝐤₃₁𝐁^′₁𝐤₄₁𝐁^′₁+𝐛⋅𝐚_ℎ21𝐛^ℎ₃₁𝐛^ℎ₄₁⋅𝐚=𝐆+𝐤𝐁_1𝑚⋅𝐚+𝐛⋅𝐚, (15) where 𝐤, 𝐁_1𝑚, and 𝐛 can be expressed as 𝐤=𝐤₂₁𝐤₃₁𝐤_413×9,𝐁_1𝑚=𝐁^′₁𝐎_3×1𝐎_3×1𝐎_3×1𝐁^′₁𝐎_3×1𝐎_3×1𝐎_3×1𝐁^′_19×3,𝐛=𝐛^ℎ₂₁𝐛^ℎ₃₁𝐛^ℎ₄₁. (16)

The vehicle magnetic interference 𝐃 is the measurement error, and it can be expressed by 𝐃=𝐤𝐁_1𝑚⋅𝐚+𝐛⋅𝐚. (17)

The actual data of the magnetic gradient tensor can be written as 𝐆=𝐆_𝑚−𝐃=𝐆_𝑚−𝐤𝐁_1𝑚⋅𝐚−𝐛⋅𝐚. (18)

Equation (18) is the compensation algorithm of the vehicle magnetic interference. 𝐤 and 𝐛 are compensation coefficient matrices, and 𝐆_𝑚, 𝐁_1𝑚 are the output of the magnetic gradiometer. In theory, (18) can be used to compensate the vehicle magnetic interference after the estimation of compensation coefficient matrices 𝐤 and 𝐛.

5. Estimation of Compensation Coefficient Matrices

The magnetic gradient tensor 𝐆 after the compensation of magnetic interference should satisfy (2). The relation of the components of 𝐆_𝑚 and 𝐃 can be obtained from the following: 𝐺_𝑚12−𝐷₁₂=𝐺_𝑚21−𝐷₂₁,𝐺_𝑚13−𝐷₁₃=𝐺_𝑚31−𝐷₃₁,𝐺_𝑚23−𝐷₂₃=𝐺_𝑚32−𝐷₃₂,𝐺_𝑚11+𝐺_𝑚22+𝐺_𝑚33=𝐷₁₁+𝐷₂₂+𝐷₃₃. (19)

We denote 𝑓(𝐤,𝐛)=𝐺_𝑚12−𝐺_𝑚21−𝐷₁₂−𝐷₂₁²+𝐺_𝑚13−𝐺_𝑚31−𝐷₁₃−𝐷₃₁²+𝐺_𝑚23−𝐺_𝑚32−𝐷₂₃−𝐷₃₂²+𝐺_𝑚11+𝐺_𝑚22+𝐺_𝑚33−𝐷₁₁+𝐷₂₂+𝐷₃₃². (20)

The estimate of the compensation coefficient matrices 𝐤 and 𝐛 can be taken as the function optimization, and it can be expressed as min𝐹(𝐤,𝐛)=min^𝑚_𝑖=1𝑓_𝑖(𝐤,𝐛),s.t.:𝑘^𝑠_𝑖𝑗≤𝑘_𝑖𝑗≤𝑘^𝑡_𝑖𝑗𝑏^𝑠_𝑖𝑗≤𝑏_𝑖𝑗≤𝑏^𝑡_𝑖𝑗. (21)

The value range of the components of compensation coefficient matrices 𝐤 and 𝐛 of ordinary vehicles is 0≤𝑘_𝑖𝑗≤0.01, 0≤𝑏_𝑖𝑗≤10. This paper proposed Genetic Algorithms (GA) method to calculate the function optimization, and the parameters of GA are chosen as follows: group size 𝑁=30, the gene length of the variate 𝑛=10, the coping probability 𝑝_𝑟=0.9, the crossover probability 𝑝_𝑐=0.8, and the mutation probability 𝑝_𝑚=0.8.

6. Numerical Simulations

6.1. Coefficients of Vehicle Magnetic Interference

Suppose that the vehicle carries a trirectangular tetrahedron magnetic gradiometer and the induced coefficient matrix 𝐊_𝑖 and the permanent magnetic field 𝐁^ℎ_𝑖 of the vector magnetometers of magnetic gradiometer are chosen as [13] 𝐊₁=,𝐊0.50.020.030.020.10.0050.030.0050.2₂=,𝐊0.5010.02020.03010.02020.10.0050.03010.0050.2₃=,𝐊0.50.02020.030.02020.1010.00510.030.00510.2₄=,𝐁0.50.020.03030.020.10.00510.03030.00510.202_ℎ1=()𝐁400025001800nT,_ℎ2=()𝐁401025001800nT,_ℎ3=()𝐁400324921800nT,_ℎ4=(400025021806)nT. (22)

6.2. Simulations Results

The magnetic gradiometer is used to measure the field of magnetic target which can be considered as magnetic dipole. The process of simulations is as follows.(1)The origin values of magnetic gradient tensor are simulated by the mathematics model of magnetic dipole.

(2)Based on the mathematics model of vehicle magnetic interference, 150 groups of 𝐆_𝑚𝑖 and 𝐁^′_1𝑖(𝑖=1,2,…,150) are simulated.

(3)Based on the estimation method of compensation coefficients, the compensation coefficient matrices 𝐤 and 𝐛 are estimated by 50 groups of 𝐆_𝑚𝑖 and 𝐁^′_1𝑖:𝐤=,0.001810.000290.000250.000010.0002900.000020.000010.000380.0002900.000010.000290.001440.000170.0000100.000170.000250.00001000.0001700.000380.000170.00313𝐛=5.364.239.166.317.261.353.894.592.54nT. (23)(4)The proposed compensation algorithm and the compensation coefficient matrices 𝐤 and 𝐛 are used to compensate the error of the other 100 groups 𝐆_𝑚𝑖 and obtain the estimations 𝐆. The partial results of simulations are shown in Table 1.

Table 1 

The results of magnetic interference compensation simulations.

The statistics of the simulation results show that the proposed method can compensate 96.2% vehicle magnetic interference, so the compensation method is efficient for the magnetic gradiometer.

7. Conclusion

Based on the study of the mechanism of the vehicle magnetic interference, we proposed a mathematical model of vehicle magnetic interference by using the difference algorithm of the magnetic gradient tensor to fuse the magnetic interference of each vector magnetometer and proposed the compensation method of the vehicle magnetic interference and the recognition method of the compensation coefficients. This method can compensate the vehicle magnetic interference of the magnetic gradiometer theoretically. The results of simulation show that the proposed method can compensate the 96.2% vehicle magnetic interference efficiently. However, it is hard to gain high precision due to the existence of so many unknown parameters in the compensation coefficients. In future works we plan to seek for other optimization algorithms to estimate the compensation coefficients and to improve the compensation of the vehicle magnetic interference.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank J. Yuan, for technical assistance, and B. L. Geng and J. F. Gai, for discussions and help with the ocean remote sensing center.